Event detail

3-Manifold Seminar: The arithmeticity of figure-eight knot orbifolds

A knot or link $L$ in $S^3$ is called universal if every closed orientable $3$-manifold can be represented as a cover branched over $L$, with some examples including the Borromean rings, the figure-eight knot, and 2-bridge non-torus links. Some such $L$ are the singular locus of an orbifold ${\mathbb H}^3/\Gamma \cong S^3$ for Γ an arithmetic subgroup of the linear algebraic group of isometries of ${\mathbb H}^3$, which gives every closed orientable manifold the structure of an arithmetic orbifold. A natural question, then, is which hyperbolic orbifolds have an arithmetic orbifold group. We will discuss a paper of Hilden-Lozano-Montesinos showing that the orbifold from the $n$-fold cyclic branched cover of the figure-eight knot is arithmetic only at the values $n=4,5,6,8,12,\infty $.